Georgia Institute of Technology College of Sciences

Nonlinear dispersive and wave equations constitute an area of PDE that has witnessed tremendous activity over the past thirty years. Such equations mostly orginate from physics; examples include nonlinear Schroedinger, wave, Klein-Gordon, and water wave equations, as well as Einstein's equations in general relativity. The rapid developments in this theory were, to a large extent, driven by several successful interactions with other areas of mathematics, mainly harmonic analysis, but also geometry, mathematical physics, probability, and even analytic number theory (we will touch on this in another talk). This led to many elegant tools and rather beautiful mathematical arguments. We will try to give a panoramic, yet very selective, survey of this rich topic focusing on intuition rather than technicalities. This first talk will deal with some aspects of nonlinear dispersive equations posed on Euclidean spaces.